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The Age of Einstein

Frank W. K. Firk

Professor Emeritus of Physics
Yale University

2003

CONTENTS

Preface 5

1. Introduction 7

2. Understanding the Physical Universe 8

3. Describing Everyday Motion 10

4. Einstein's Theory of Special Relativity 18

5. Newton's Dynamics 36

6. Equivalence of Mass and Energy: E = mc 2 45

7. An Introduction to Einstein's General Relativity 50

8. Appendix: a Mathematical Approach to Special Relativity 62

9. Bibliography 82

PREFACE

This book had its origin in a one-year course that I taught at Yale
throughout the decade of the 1970's. The course was for non-science majors
who were interested in learning about the major branches of Physics. In the
first semester, emphasis was placed on Newtonian and Einsteinian Relativity.
The recent popularity of the Einstein exhibit at the American Museum of
Natural History in New York City, prompted me to look again at my fading
lecture notes. I found that they contained material that might be of interest to
today's readers. I have therefore reproduced them with some additions, mostly
of a graphical nature. I recall the books that were most influential in my
approach to the subject at that time; they were Max Born's The Special Theory
of Relativity, Robert Adair's Concepts of Physics, and Casper and Noer's
Revolutions in Physics. These three books were written with the non- scientist
in mind, and they showed what could be achieved in this important area of
teaching and learning; I am greatly indebted to these authors.

1. INTRODUCTION

This brief book is for the inquisitive reader who wishes to gain an understanding of the
immortal work of Einstein, the greatest scientist since Newton. The concepts that form the
basis of Einstein's Theory of Special Relativity are discussed at a level suitable for Seniors in
High School. Special Relativity deals with measurements of space, time and motion in inertial
frames of reference (see chapter 4). An introduction to Einstein's Theory of General Relativity,
a theory of space, time, and motion in the presence of gravity, is given at a popular level. A
more formal account of Special Relativity, that requires a higher level of understanding of
Mathematics, is given in an Appendix.

Historians in the future will, no doubt, choose a phrase that best characterizes the 20th-
century. Several possible phrases, such as "the Atomic Age", "the Space Age" and "the
Information Age", come to mind. I believe that a strong case will be made for the phrase "the
Age of Einstein"; no other person in the 20th-century advanced our understanding of the
physical universe in such a dramatic way. He introduced many original concepts, each one of
a profound nature. His discovery of the universal equivalence of energy and mass has had, and
continues to have, far-reaching consequences not only in Science and Technology but also in
fields as diverse as World Politics, Economics, and Philosophy.

The topics covered include:

a) understanding the physical universe;

b) describing everyday motion;

relative motion,

Newton's Principle of Relativity,

problems with light,

c) Einstein's Theory of Special Relativity;

simultaneity and synchronizing clocks,
length contraction and time dilation,
examples of Einstein's world,

d) Newtonian and Einsteinian mass;

e) equivalence of energy and mass, E^mc 2 ;

f) Principle of Equivalence;

g) Einsteinian gravity;

gravity and the bending of light,

gravity and the flow of time, and

red shifts, blue shifts, and black holes.
2. UNDERSTANDING THE PHYSIC AL UNIVERSE

We would be justified in thinking that any attempts to derive a small set of fundamental
laws of Nature from a limited sample of all possible processes in the physical universe, would

lead to a large set of unrelated facts. Remarkably, however, very few fundamental laws of
Nature have been found to be necessary to account for all observations of basic physical
phenomena. These phenomena range in scale from the motions of minute subatomic systems
to the motions of the galaxies. The methods used, over the past five hundred years, to find the
set of fundamental laws of Nature are clearly important; a random approach to the problem
would have been of no use whatsoever. In the first place, it is necessary for the scientist to have
a conviction that Nature can be understood in terms of a small set of fundamental laws, and
that these laws should provide a quantitative account of all basic physical processes. It is
axiomatic that the laws hold throughout the universe. In this respect, the methods of Physics
belong to Philosophy. (In earlier times, Physics was referred to by the appropriate title,
"Natural Philosophy").
2.1 Reality and Pure Thought

In one of his writings entitled "On the Method of Theoretical Physics", Einstein stated:
"If, then, experience is the alpha and the omega of all our knowledge of reality , what then is the
function of pure reason in science?" He continued, "Newton, the first creator of a
comprehensive, workable system of theoretical physics, still believed that the basic concepts
and laws of his system could be derived from experience." Einstein then wrote "But the
tremendous practical success of his (Newton's) doctrines may well have prevented him, and
the physicists of the eighteenth and nineteenth centuries, from recognizing the fictitious
character of the foundations of his system". It was Einstein's view that "..the concepts and

fundamental principles which underlie a theoretical system of physics am free inventions of the
human intellect, which cannot be justified either by the nature of that intellect or in any other
fashion a priori^ He continued, "If, then, it is true that the axiomatic basis of theoretical
physics cannot be extracted from experience but must be freely invented, can we ever hope to
find the right way? ... Can we hope to be guided safely by experience at all when there exist
theories (such as Classical (Newtonian) Mechanics) which to a large extent do justice to
experience, without getting to the root of the matter? I answer without hesitation that there is, in
my opinion, a right way, and that we are capable of finding it." Einstein then stated
"Experience remains, of course, the sole criterion of the physical utility of a mathematical
construction. But the creative principle resides in Mathematics. ... I hold it true that pure
thought can grasp reality, as the ancients dreamed."
3. DESCRIBING EVERYDAY MOTION
3.1 Motion in a straight line (the absence of forces)

The simplest motion is that of a point, P, moving in a straight line. Let the line be
labeled the "x-axis", and let the position of P be measured from a fixed point on the line, the
origin, O. Let the motion begin (time t = 0) when P is at the origin (x = 0). At an arbitrary time,
t, P is at the distance x:

P[x,t]

-x O position, x +x

at timet
If successive positions of P are plotted, together with their corresponding times, we can
generate what is called the "world line" of P.

Let us observe a racing car moving at high speed along the straight part of a race track
(the x-axis) , and let us signal the instant that it passes our position, x = 0, by lowering a flag:
An observer, standing at a measured distance D, from x = 0, starts his clock at the instant, t = 0,
when he sees the flag lowered, and stops his clock at the instant t = T, as the car passes by.
We can obtain the average speed of the car, v, during the interval T, in the standard way; it is

v = D/T (in units of velocity) .
If, for example, D = 1 mile, and T = 20 seconds (1/180 hour), then
v = 1 (mile)/( 1/180) (hour) =180 miles per hour.

This is such a standard procedure that we have no doubt concerning the validity of the result.
32 The relativity of everyday events

Events, the description of when and where happenings occur, are part of the physical
world; they involve finite extensions in both time and space. From the point of view of a
theory of motion, it is useful to consider "point-like" events that have vanishingly small
extensions in time and space. They then can be represented as "points" in a space-time

geometry. We shall label events by giving the time and space coordinates: event E -> E[t, x] ,
or in three space dimensions, E[t, x, y, z], where x, y, z are the Cartesian components of the
position of the event. There is nothing special about a Cartesian coordinate system, it is a
mathematical construct; any suitable coordinate mesh with a metrical property (measured
distances defined in terms of coordinates) can be used to describe the spatial locations of
events . A familiar non-Cartesian system is the spherical polar coordinate system of the lines of
latitude and longitude on the surface of the earth. The time t can be given by any device that is
capable of producing a stable, repetitive motion such as a pendulum, or a quartz-controlled
crystal oscillator or, for high precision, an atomic clock.

Suppose we have an observer, O, at rest at the origin of an x-axis, in the F-frame. O
has assistants with measuring rods and clocks to record events occurring on the x-axis:

-x O +X

We introduce a second observer, O', at rest at the origin of his frame of reference, F. O' has
his assistants with their measuring rods (to measure distances, x 7 ) and clocks (to measure times,
t') to record events on the x'-axis. (The F-clocks are identical in construction and performance
to the clocks in the F-frame). Let O' coincide with O at a common origin O = O' (x = x' = 0), at
the synchronized time zero t = t 1 = 0. At t = t'= 0, we have

< =-- ►

-x' O' +x'

13
Suppose that the observer O', and his assistants with rods and clocks, move to the right with

constant speed V along the common x, x -axis. At some later time t, the two sets of observers ,

represented by O and O', record a common event that they write as E[t, x] and E'[t', x'],

respectively. The relative positions of the two observers at time t is:

•* —

E[t.x]

— • ►

-X

0^ x -►

F=^ E'[t',

x']

-4

it — — k

-x'

<y — x'- -*

+x'

where D = Vt is the distance that O' moves at constant speed V, in the time t.

We therefore write the relationship between the two measurements by the plausible equations

(based on everyday experience):

t' = t (everyday identical clocks tick at the same rate)

and

x' = x-D = x-Vt.

These are the basic equations of relative motion according to the concepts first put forward by
Galileo and Newton. They are fully consistent with measurements made in ourreal world (the
world of experience) . They are not, however, internally consistent. In the equation that relates

the measurement of distance x' in the F-frame to the measurements in the F-frame, we see that

the space part, x', in the F-frame, is related to the space part, x, and the time part, t, in the F-
frame: space-time in one frame is not related to space-time in the other frame! Furthermore,
the time equation makes no mention of space in either frame. We see that there is a
fundamental lack of symmetry in the equations of relative motion, based on everyday
experience. The question of the "symmetry of space-time" will lead us to Einstein's
philosophy of the "free invention of the intellect'.
33 Relative velocities

We have seen that the position of an event, E[t, x], measured by an observer O, is
related to the position of the same event, E'[t', x'], measured by an observer O', moving with
constant speed V along the common x, x'-axis of the two frames, by the equation
x'=x-Vt.

The speed v of a point P[t , x] , moving along the x-axis , is given by the ratio of the finite
distance the point moves, Ax, in a given finite time interval, At:

v= Ax/At.
We can obtain the speeds v, and V of the same moving point, as measured in the two frames,
by calculating v = Ax/At and v' = Ax'/At', as follows:

Ax'/At' = V = Ax/At - VAt/At (where we have used At' = At because t = f in everyday
experience).
We therefore find

v' = v-V,

the speeds differ by the relative speed of the two frames . This is consistent with experience: if

a car moves along a straight road at a constant speed of v = 60 mph, relative to a stationary

observer O, and an observer O' follows in a car at a constant speed of V = 40 mph relative to

O, then the speed of the first car relative to the occupant of the second car is V = 20 mph.

34 The Newtonian Principle of Relativity

The Newtonian Principle of Relativity asserts that, in the inertial frames F, F, the

following two situations

x=x'=0att=t'=0
i

-x <— O ^+x

D = Vt

•* ►

+V F (moves to right
at speed V)

-N-x'

D' = Vf F

•4 ►

-V <= F (moves to left at speed V)

Q^ -*+x

cannot be distinguished by experiments that involve mechanical systems (classical systems
that obey Newton's Laws of Motion).

The speed V has been written in bold face to remind us that here we are dealing with a
vector quantity that has both magnitude (the speed in mph) and a sense of direction: +V in the
+x-direction and -V in the -x-direction.
35 Problems with light

We are accustomed to the notion that waves propagate through a medium, required to
support the waves. For example, sound waves propagate as pressure variations in air, and water
waves propagate as coupled displacements of the water molecules, perpendicular to the
direction of the wave motion. In the 19th-century, Maxwell discovered that light waves are
electromagnetic phenomena. This great work was based on theoretical arguments, motivated
by the experimental results of Faraday and Henry. One of the most pressing questions facing
scientists of the day was:

"what is waving when a beam of light propagates through empty space?"
It was proposed that the universe is filled with a medium called the aether with the property of
supporting light waves, and having no other physical attributes. (For example, it would have
no effect on the motion of celestial bodies). In the latter part of the 19th-century, Michelson
and Morley carried out a famous experiment at the Case Institute in Cleveland that showed
there is no experimental evidence for the aether. Light travels through the void, and that is that.
Implicit in their work was the counter-intuitive notion that the speed of light does not depend
on the speed of the source of the light.

17
The Aether Theory was popular for many years. Non-traditional theories were
proposed to account for the null-result of the Michelson-Morley experiment. Fitzgerald
(Trinity College, Dublin) proposed that the Michelson-Morley result could be explained, and
the Aether Hypothesis retained, if the lengths of components in their apparatus were "velocity-
dependent" - lengths contract in the direction of motion, and lengths remain unchanged when
perpendicular to the direction of motion. He obtained the result
L„ = [W(l-(v/cf)]L m yh

ft It

(length of rod at rest) (length of rod moving at speed v)

Here, c is the constant speed of light (2.99 . . . x 10 8 meters/second).

All experiments are consistent with the statement that the ratio v/c is always less than 1, and

therefore y is always greater than 1 . This means that the measured length of the rod Lq , in its

rest frame, is always greater than its measured length when moving.

At the end of the 19th-century, Larmor introduced yet another radical idea: a moving

clock is observed to tick more slowly than an identical clock at rest. Furthermore, the

relationship between the clock rates in the moving and rest frames is given by the same factor,

y, introduced previously by Fitzgerald. Specifically,

At = Y At o

It It

(an interval on a moving clock) (an interval on a clock at rest)

Since the velocity-dependent term y is greater than 1, the intervals of time At (moving), are

greater than the intervals Ato (at rest).

Fitzgerald, Larmor, and other physicists at that time considered length contraction and
time dilation to be "real" effects, associated with minute physical changes in the structure of
rods and clocks when in motion. It was left to the young Einstein, working as a junior Patent
Officer in Bern, and thinking about space, time, and motion in his spare time, to introduce a
new Theory of Relativity, uninfluenced by the current ideas. Although, in later life, Einstein
acknowledged that he was aware of the Michelson-Morley result, and of the earlier work on
length contraction and time dilation, he neither used, nor refened to, the earlier results in his first
paper on Relativity.
4. EINSTEIN'S THEORY OF SPECIAL RELATIVITY

In 1905, Einstein published three great papers in unrelated areas of Physics. In this
chapter, we shall discuss his new ideas concerning the relative motion of beams of light, and of
objects that move at speeds close to that of light. His independent investigations were based
upon just two postulates:

1. The generalized Principle of Relativity: no mechanical or optical experiments can
be carried out that distinguish one inertial frame of reference from another. (This is a
development of Newton's Principle of Relativity that is limited to mechanical experiments,
involving speeds much less than the speed of light; it applies to experiments in the everyday
world).

19
Inertial frames of reference are non-rotating, and move in straight lines at constant
speed. They are non-accelerating.

2. The speed of light in a vacuum is a constant of Nature, and is independent of the
velocity of the source of the light.

Einstein was not concerned with questions having to do with the Aether; for him, a true theory
of the physical properties of the universe could not rest upon the mysterious qualities of such an
unobservable. As we shall see, Einstein was concerned with the precise meaning of
measurements of lengths and time intervals. In his later years, he recalled an interesting
thought that he had while in school. It had to do with the meaning of time. Our lives are
dominated by "psychological time"; for example, time seems to go by more quickly as we
grow older. For the young Einstein, time in the physical world was simply the reading on a
clock. He therefore imagined the following: if the schoolroom clock is reading 3PM, and I
rush away from the clock at the speed of light, then the information (that travels at the speed of
light) showing successive ticks on the clock, and therefore the passage of time, will never reach
me, and therefore, in my frame of reference, it is forever 3PM - time stands still. He therefore
concluded that the measurement of time must depend, in some way, on the relative motion of
the clock and the observer; he was, by any standards, a precocious lad.

If we apply the Galilean-Newtonian expression for the relative velocities v, V,
measured in the inertial frames F, F, (moving with relative speed V), to the measurements of
flashes of light, v = c, the speed of light in F, and V = c', the speed of light in F, we expect

c' = c - V (corresponding to V = v - V for everyday objects).
Modern experiments in Atomic, Nuclear, and Particle Physics are consistent with the fact that
c'=c,no matter what the value of the relative speed V happens to be.

At the end of the 19th-century, a key question that required an answer was therefore: -
why does the Galilean-Newtonian equation, that correctly describes the relative motion of
everyday objects, fail to describe the relative motion of beams of light? Einstein solved the
problem in a unique way that involved a fundamental change in our understanding of the
nature of space and time, a change that resulted in far-reaching consequences; these
consequences are discussed in the following chapters.
4.1 The relativity of simultaneity: the synchronization of clocks.

It is important to understand the meaning of the word "observer" in Relativity. To
record the time and place of a sequence of events in a particular inertial reference frame, it is
necessary to introduce an infinite set of adjacent "observers", located throughout the entire
space. Each observer, at a known, fixed position in the reference frame, carries a clock to
record the time, and the characteristic property, of every event in his immediate neighborhood.
The observers are not concerned with non-local events. The clocks carried by the observers
are synchronized: they all read the same time throughout the reference frame. It is the job of the
chief observer to collect the information concerning the time, place, and characteristic feature
of the events recorded by all observers, and to construct the world line (a path in space-time),
associated with a particular characteristic feature (the type of particle, for example). "Observer"

21
is therefore seen to be a collective noun, representing the infinite set of synchronized observers
in a frame of reference.

The clocks of all observers in a reference frame are synchronized by correcting them
for the speed of light (the speed of information) as follows:

Consider a set of clocks located at Xq, x 1? x 2 , x 3 , along the x-axis of a reference frame.
Let Xq be the position of the chief observer, and let a flash of light be sent from the clock at Xq
when it reads ^ ( 12 noon, say) . At the instant that the light signal reaches the clock at Xj , it is set
to read ^ + (x/c), at the instant that the light signal reaches the clock at %, it is set to read \q +
(x^c) , and so on for every clock along the x-axis. All clocks along the x - axis then "read the
same time" - they are synchronized:

These 4 clocks read the same time "noon + xjc in their rest frame

du. ®~~> ®—> o~>

Xq Xj X 2 X 3

noon at Xq noon+x/c noon + Xj/c noon + x/c

To all other inertial observers, the clocks appear to be unsynchronized.

The relativity of simultaneity is clearly seen using the following method to synchronize
two clocks: a flash of light is sent out from a source, M situated midway between identical
clocks, A and B' , at rest in the frame, F

A! v M' B'

Flash of light from M , the mid-point between A' and B '

The two clocks are synchronized by the (simultaneous) arrival of the flash of light (traveling at
c) from M\ We now consider this process from the viewpoint of observers in an inertial
frame, F, who observe the F-frame to be moving to the right with constant speed V. From
their frame, the synchronizing flash reaches A' before it reaches B':

F'V original position of M'

A^cV

cB^ V

*-Jt

23
They conclude, therefore, that the A'-clock starts before the B'-clock; the clocks are no longer
synchronized. This analysis rests on the fact that the speed of light does not depend on the
speed of the source of light.

The relativity of simultaneity leads to two important non-intuitive results namely,
length contraction and time dilation.
42 Length contraction
Let a rod be at rest in the F-frame, and let its proper (rest) length be Lq.

L„

Consider an F-frame, moving at constant speed V in the +x direction. The set of observers, at

rest in F , have synchronized clocks in F , as shown

->V All clocks "read the same time" in F

The observers in F wish to determine the length of the rod, L, relative to the F frame. From
their perspective, the rod is moving to the left (the -x' direction) with constant speed, V. We

define the length of a rod, measured in any inertia! frame, in terms of the positions of the two

ends of the rod measured at the same time. If the rod is at rest, it does not matter when the two
end positions are determined; this is clearly not the case when the rod is in motion. The
observers in F are distributed along the x -axis, as shown. They are told to measure the length
of the rod at 12 noon. This means that, as the rod passes by, each observer looks to see if either
end of the rod is in his (immediate) vicinity. If it is, the two critical observers A' and B' (say)
raise their hands. At any time later, the observers in F measure the distance between the
observers A and B', and the chief observer states that this is the length of the rod, in their frame.
This procedure can be carried out only if the clocks in F are synchronized. We have seen,
however, that the synchronized clocks in F are not seen to be synchronized in a different
frame, F, such that F moves at speed V, relative to F. The question is: how does the length L'
of the moving rod, determined by the observers in F, appear to the observers at rest in the F
frame? We have seen that the clock A starts before clock B', according to the F observers.
Therefore "left end of the rod coincides with the A clock, reading noon" occurs before "right
end of rod coincides with B' clock, reading noon", according to the F observers:

F F V position of left end at t" = noon

►

A'(D/ ^)B^ notyett' = noon

B' an intermediate position

F V position of right end at t 1 = noon

A'Y L' B'V

according to observers in F

Lo,inF

The length of the rod, L', as determined by observers at rest in F, shown from the viewpoint of
observers in the F frame, is less than its proper length, L,,.

We see that the measurement of length contraction does not involve physical changes
in a moving rod; it is simply a consequence of the synchronization of clocks in inertial frames,
and the relativity of simultaneity.

43 Time dilation

The proper time interval between two events is the interval measured in the frame of
reference in which they occur at the same position. Intervals that take place at different
positions are said to be improper.

Consider a pulse of light that reflects between two plane mirrors, Mj and M 2 , separated
by a distance D:

Pulse of light traveling between M l and M 2 ,
at the invariant speed of light, c

The time interval, At, for the light to make the round-trip is At = 2D/c.

Consider a reference frame, F, (the laboratory frame, say), and let the origin of F coincide with

the location of an event Ej . A second event F^, occurs at a different time and location in F, thus

E,
t

1 st event atx = t=0inF 2nd event at a different place and different time in F

27
Let us introduce a second inertial frame, F, moving with speed V relative to F, in the +x
direction:

2nd event, at the "same place", O , in F

O'O

Let F be chosen in such a way that its origin, O' , coincides with the first event E, (in both space
and time), and let its speed V be chosen so that O coincides with the location of the second
event, L^. We then have the situation in which both events occur at the someplace in F'(at the
origin, O'). The interval between E, and L^ in F is therefore ^proper interval.
The minors, M, and M 2 , are at rest in F, with M, at the origin, O'. These mirrors move to the
right with speed V. Let a pulse of light be sent from the lower mirror when O and O coincide
(at the instant that E l occurs). Furthermore, let the distance D between the mirrors be adjusted
so that the pulse returns to the lower mirror at the exact time and place of the event, L^. This

sequence of events, as observed in F, is as shown:

►

M 2 ,

F \v

\/r \

Distance =

= VAL \

Event Ej ,
light leaves Mj

The sequence, observed in F is

At time At/2, later,

light reflects from M 2

Event F^
light arrives at M,

at time At

D The mirrors are at rest in F , and the events
Ej and F^ occur at the someplace, therefore
The time interval At' = 2D/c is aproper interval.

Ej and E^ both take place at the origin, O'

The geometry of the sequence of events in the F frame is Pythagorean.

Recall, that in a right-angled triangle, we have

a 2 +b 2 = c 2 .

In F, the relevant distances are:

VAt/2

-> -4-

VAt/2

where At is the round-trip travel time. We therefore have

(VAt/2) 2 + D 2 = (cAt/2) 2 ,

or,

(At) 2 [c 2 -V 2 ]=4D 2 ,
so that, on taking the square root,

At=2D//(c 2 - V 2 ) = {lA/Tl - (V/c) 2 ]}(2D/c) =y(2D/c),

where y is the factor first introduced, on empirical grounds, by Fitzgerald. In this discussion, we

see that it emerges in a natural way from the two postulates of Einstein.

We can now compare the interval, At, between E l and F^ in F with the interval, At', between
the same events as determined in F. If we look at the sequence of events in F, in which the
mirrors are at rest, we have

2D = cAt\
and therefore

At' = 2D/c.
Substituting this value in the value for At in F, we obtain

At = {l//[l-(V/c) 2 ]}Af
or

At=yAt',Y>l,
which means that At (moving) > At' (at rest).
A moving clock runs more slowly than an equivalent clock at rest.

Notice that at everyday speeds, in which V/c is typically less than 10 6 , (and therefore (V/c) 2 is
less than 10" 12 , an unimaginably small number), At and At' are essentially the same. Einstein's
result then reduces to the classical result of Newton. However, in Modem Physics, involving
microscopic particles that have measured speeds approaching that of light, values of y > 1000
are often encountered. The equations of Newtonian Physics, and the philosophical basis of the
equations, are then fundamentally wrong.

Although we have used an "optical clock" in the present discussion, the result applies to clocks
in general, and, of course, to all inertia! frames (they are equivalent).

31
A formal discussion of Einstein's Theory of Special Relativity is given in the
Appendix.; it is intended for those with a flair for Mathematics.
44 Experimental evidence for time dilation and length contraction

At the top of the Earth's atmosphere, typically 30,000m above sea-level, various gasses
are found, including oxygen. When oxygen nuclei are bombarded with very high energy
protons from the Sun, and from more distant objects, entities called muons are sometimes
produced. These muons are found to have speeds very close to that of light (> 0.999c).
Experiments show that the life-time of the muon, in its rest frame, is very short, a mere 2 x 10 "
6 seconds. After that brief existence, the muon transforms into other elementary particles. In
Newtonian Physics, we would therefore expect the muon to travel a distance d = VAt, where V
« c * 3 x 10 8 m/s, and At = 2 x 10 " 6 s, so that d « 600 m. We should therefore never expect to
observe muons on the surface of the Earth, 30,000 m below. They are, however, frequently
observed here on Earth, passing through us as part of the general cosmic background.
Although the lifetime of the muon is 2^is in its rest frame, in the rest frame of the Earth, it is
moving very rapidly, and therefore the interval between its creation and decay is no longer At
but rather

Atj, = yAt^ = 2 x 10 - 6 seconds / V [1 - (V/c) 2 ]
where V is the speed of the muon relative to Earth, and At^ is the lifetime of the muon in its rest
frame. Atg is its lifetime in the Earth's frame.

We see that if y s 50, the muon will reach the Earth. (For then, Atg > 10 ~ 4 s , and therefore Hg

> 30,000m). A value of y = 50 corresponds to a muon speed V = 0.9995c, and this is

consistent with observations.

The detection ofmuons on the surface of the Earth is direct evidence for time dilation.

Alternatively, we may consider the observation of muons at the Earth's surface in terms of

length contraction, as follows:

If we view the Earth from the rest frame of the muon, 30,000m above the Earth, it is moving

toward the muon with very high speed, V corresponding to a value y > 50. According to

Einstein, the distance to the Earth is contracted by a factor of y > 50, so that the muon-Earth

distance from the perspective of the muon is FT = 30000/y < 30000/50 < 600m. The Earth

therefore reaches the muon before it decays.

This is direct evidence for length contraction.

Schematically, we have:

muon at nest in F. , seen to

" ^be noshing tow ard Earth

H E = 30km

Earth at rest in F p

muon at rest in F,„

Earth rushing up

H u <600m

In F E , fixed to the Earth, a muon

In F , fixed to the muon,

moving at high speed V toward

the Earth is moving

the Earth, is created at H E = 30km.

upwards at very high speed, V,

at a contracted distance

H (X =V(l-(V/c) 2 )H E .

= H E / 50 for V = 0.9995c.

45 Space travel

The distance between any two stars is so great that it is measured in "light-years", the
distance light travels in one year. In more familiar units:

1 light-year = 9.45 x 10 15 meters * 6 trillion miles!

Alpha Centauri, our nearest star, is 4.3 light-years away; this means that, even if a
spaceship could travel at a speed close to c, it would take more than 4.3 years to reach the star.

Imagine that it were technically possible to build such a near-light-speed craft; we ask
how long the journey would take to a star, 80 light-years away, as measured by observers on
the Earth. Let E^ be the event "craft leaves Earth" and Ej^ the event "craft arrives at star".
To observers in the rest frame of the Earth, it travels a distance 756 x 10 15 meters at a speed of
0.995c, (say), in a time
Atj, = 756 x 10 15 (meters) / 0.995 x 9.45 x 10 15 (meters per light-year)
= 80.4 years.
Very few passengers would be alive when the craft reached the star. If the survivors sent a
radio signal back to the Earth, saying that they had arrived, it would take another 80 years for
the signal to reach the Earth. (Radio waves travel at the invariant speed of light).

Let us calculate the time of the trip from the perspective of the travelers. The
spacecraft-fixed frame is labeled E^ In this frame, the two events, E^ and E^^ occur at the
same place, namely the origin of F^. The trip-time, At^, according to the travelers is a

35
"proper" time interval, and is less than the trip-time according to Earth-fixed observers by a
factor vtl-CWc) 2 ), where V = 0.995c. We then find
At^ = At/Cl - (0.995) 2 ) = Atg x v^GAl)
= 80.4x0.1
= 8.04 years.
To the travelers on board, the trip takes a little more than 8 years.

The passengers and crew of the spacecraft would have plenty of time to explore their new
environment. Their return trip would take another 8.04 years, and therefore, on arrival back on
the Earth, they would have aged 16.08 years, whereas their generation of Earth-bound persons
would have long-since died. (The society would have aged 160 years).

This discussion assumes that the biological processes of the travelers take place
according to time on the spacecraft clocks. This is certainly reasonable because heart-beats
represent crude clocks, and metabolic rates of life processes are clock-like. According to
Einstein, all "clocks" are affected by the motion of one inertial frame relative to another.

The twin paradox is not a paradox at all. Consider a twin, A, in the rest frame of the
Earth, and let A observe the round-trip of his twin, B, to a distant star at near-c speed. A
concludes that when B returns to Earth, they are no longer the same age; B is younger than A.
According to the Principle of Relativity, B can be considered at rest, and the Earth, and twin A,
travel away and return, later. In this case, B concludes that A has been on the trip, and
therefore comes back younger. This cannot be ! The paradox stems from the fact that the twin,

B, who leaves the Earth, must accelerate from the Earth, must slow down at the star, turn
around, accelerate away from the star, and slow down on reaching Earth. In so doing, the B-
twin has shifted out of inertia! frames and into accelerating frames. Special Relativity does not
hold throughout the entire journey of the twin in the spacecraft. The A-twin is always at rest in
the inertial frame of the Earth. There is ^permanent asymmetry in the space-time behavior of
the twins.

The time in which a space traveler is in non-inertial frames can be made very short
compared with the total travel time. The principles of Special Relativity are then valid, and the
discussion given above, in which a spacecraft travels to a distant star, 80 light-years away, is
essentially correct.
5. NEWTONIAN DYNAMICS

. . .for the whole burden of (natural) philosophy seems to consist of this
—from the phenomena of motions to investigate the forces of nature, and then from these
forces to demonstrate the other phenomena.

NEWTON, the PRINCIPIA

Although our discussion of the geometry of motion has led to major advances in our
understanding of measurements of space and time in different inertial systems, we have yet to
come to the real crux of the matter, namely, a discussion of the effects of forces on the motion
of two or more interacting particles. This key branch of Physics is called Dynamics. It was
founded by Galileo and Newton and perfected by their followers, notably Lagrange and

37
Hamilton. The Newtonian concepts of mass, momentum and kinetic energy require
fundamental revisions in the light of the Theory of Special Relativity. In spite of the conceptual
difficulties inherent in the classical theory, its success in accounting for the dynamical behavior
of systems, ranging from collisions of gas molecules to the motions of planets has been, and
remains, spectacular.
5.1 The law of inertia

Galileo (1564-1642) was the first to develop a quantitative approach to the study of
motion of everyday objects. In addition to this fundamental work, he constructed one of the
first telescopes and used it to study our planetary system. His observation of the moons of
Jupiter gave man his first glimpse of a miniature world system that confirmed the concepts put
forward previously by Copernicus (1473 -1543).

Galileo set out to answer the question: what property of motion is related to force? Is it
the position of the moving object or its velocity or its rate of change of velocity, or what? The
answer to this question can only be obtained from observations, this is a basic feature of
Physics that sets it apart from Philosophy proper. Galileo observed that force Influences
changes in velocity (accelerations) of an object and that, in the absence of external forces (e.g.
friction), no force is needed to keep an object in motion that is traveling in a straight line with
constant speed. This observationally based law is called the Law of Inertia. It is, perhaps,
difficult for us to appreciate the impact of Galileo's new ideas concerning motion. The fact that
an object resting on a horizontal surface remains at rest unless something we call force is

applied to change its state of rest was, of course, well-known before Galileo's time. However,
the fact that the object continues to move after the force ceases to be applied caused
considerable conceptual difficulties to the early Philosophers. The observation that, in practice,
an object comes to rest due to frictional forces and air resistance was recognized by Galileo to
be a side effect and not germane to the fundamental question of motion. Aristotle, for example,
believed that the true or natural state of motion is one of rest. It is instructive to consider
Aristotle's conjecture from the viewpoint of the Principle of Relativity: is a natural state of rest
consistent with this general Principle? First, we must consider what is meant by a natural state
of rest; it means that in a particular frame of reference, the object in question is stationary.
Now, according to the general Principle of Relativity, the laws of motion have the same form
in all frames of reference that move with constant speed in straight lines with respect to each
other. An observer in a reference frame moving with constant speed in a straight line with
respect to the reference frame in which the object is at rest, would conclude that the natural state
of motion of the object is one of constant speed in a straight line and not one of rest. All inertial
observers, in an infinite number of frames of reference, would come to the same conclusion.
We see, therefore, that Aristotle's conjecture is not consistent with this fundamental Principle.

39
52 Newton's laws of motion

During his early twenties, Newton postulated three laws of motion that form the basis
of Classical Dynamics. He used them to solve a wide variety of problems, including the
motion of the planets. They play a fundamental part in his famous Theory of Gravitation. The
laws of motion were first published in ihePrlncipia in 1687; they are:

1. In the absence of an applied force, an object will
remain at rest or in its present state of constant
speed in a straight line (Galileos Law of Inertia)

2. In the presence of an applied force, an object will
be accelerated in the direction of the applied force
and the product of its mass by its acceleration is
equal to the force.

and,

3. If a body A exerts aforce of magnitude IF^J on a body
B, then B exerts an equal force of magnitude IF BA I on
A. The forces act in opposite directions so that

F AB - F BA •

The 3rd. Law applies to "contact interactions". For non-contact interactions, it is necessary to
introduce the concept of a "field-of-force" that "carries the interaction".

We note that in the 2nd law, the acceleration lasts only while the applied force lasts.
The applied force need not, however, be constant in time; the law is true at all instants during
the motion. We can show this explicitly by writing:

F(t) = ma(t)
where the time-dependence of the force, and the resulting acceleration, is emphasized.

The "mass" appearing in Newton's 2nd law is the so-called inertial mass. It is that
property of matter that resists changes in the state of motion of the matter. Later, indiscussions
of Gravitation, we shall meet another property of matter that also has the name "mass" ; it is that
property of matter that responds to the gravitational force due to the presence of other
"masses"; this "mass" is the so-called "gravitational mass". The equivalence of inertial and
gravitational mass was known to Newton. Einstein considered the equivalence to be of such
fundamental importance that he used it as a starting point for his General Theory of Relativity,
one of the greatest creations of the human mind.

In 1665 - 66, Sir Isaac Newton, the supreme analytical mind to emerge from England,
deduced the basic law governing the interaction between two masses, M, and M 2 . The force
depends on tine product of the two masses, and the square of the distance between them, thus
Mass, Mi Mass,M 2

Distance between centers, R

The gravitational force between the masses is given by:

MjxM 2

F oc

giav

R 2

If the masses are initially 1 meter apart, and we increase the separation to 2
meters, the force decreases by l/(2f - 1/4.

(It took Newton many years to prove that the distributed mass of a sphere
can be treated as a "point" mass at its center. The problem involves a three-
dimensional integral using his newly-invented Calculus).
53 General features of inverse square-law forces

In the early 1700's, Coulomb deduced the law of force that governs
the interaction between two objects that possess the attribute of "electric
charge". He found the following: the force between two charges Q { (at rest)
and Q 2 depends on the product of the two charges, Q l and Q 2 , and on the
square of the distance between them:

Charge Q l Charge Q 2

Distance between centers, R
QiXQ 2

elect

R 2

In the 19th-century, experiments showed that charges in motion, relative to an observer,
generate an additional component of the force called the Magnetic Force. The complete force
between moving charges is therefore known as the Electromagnetic Force.

We see that there is a remarkable similarity between the forms of the Gravitational and
the Electromagnetic forces. They both depend on the symmetries Mj x M 2 and Q l x Q 2 , and
they both vary as the inverse square of the distance between the objects. This latter feature is
not by chance.

Let us introduce a model of these interactions in which we postulate that the force between one
object and another is "carried", or mediated, by entities, generated by their sources; let them
travel in straight lines between the objects. The mediators are capable of transferring
momentum between the interacting objects. Consider the case in which a stationary charge Q
is the source of mediators that travel, isotropically, from the charge at a uniform rate:

Charge Q

Area. A at lm.

4Aat2m.

It is a property of the space in which we live that the shaded area, A, situated 1 meter from the
charge Q, projects onto an area 4A at a distance 2 meters from Q. Therefore, the number of
mediators passing through the area A at 1 meter from Q passes through an area 4A at a
distance 2 meters from Q. If the force on a second charge Q', 1 meter from Q, is due to the
momentum per second transfened to the area A as a result of the mediators striking that area,
then a charge Q', 2 meters from the source, will experience a force that is 1/4 the force at 1
meter because the number of mediators per second passing through A now passes through an
area four times as great. We see that the famous "inverse square law" is basically geometric in
origin.

Not surprisingly, the real cases are more subtle than implied by this model. We must
recognize a fundamental difference between the gravitational and the electromagnetic forces,
namely:

the gravitational force is always attractive, whereas the electromagnetic force can be
either attractive or repulsive. This difference comes about because there is only "one kind" of
mass, whereas there are "two kinds" of charge, which we label positive and negative. (These
terms were introduced by the versatile Ben Franklin). The interaction between like charges is
repulsive and that between unlike charges is attractive.

We can develop our model of forces transmitted by the exchange of entities between
objects that repel each other in the following way:

Consider two boxes situated on a sheet of ice. Let each box contain a person and a supply of
basketballs. If the two occupants throw the balls at each other in such a way that they can be
caught, then a stationary observer, watching the exchange, would see the two boxes moving
apart. (This is a consequence of a law of motion that states that the linear momentum of a
system is conserved in the absence of external forces):

If the observer were so far away that he could see the boxes, but not the balls being exchanged,

he would conclude that an unexplained repulsive force acted between the boxes.

The exchange model of an attractive force requires more imagination; we must invoke the

exchange of boomerangs between the occupants of the

boxes, as shown

Boxes move together

These models are highly schematic. Nonetheless, they do indicate that models based
on the exchange of entities that carry momentum, can be constructed. Contemporary theories
of the Nuclear Force, and the Quark- Quark Force of Particle Physics, involve the exchange of
exotic entities (mesons and gluons, respectively).

Newton deduced the inverse square law of gravitation by combining the results of
painstaking observations of the motions of the planets (Brahe and Kepler), with an analysis of
the elliptical motion of a (terrestrial) object, based on his laws of motion. This was the first time
that the laws of motion, discovered locally, were applied on a universal scale.
6. EQUIVALENCE OF MASS AND ENERGY: E=mc 2
6.1 Relativistic mass

In Newtonian Physics, the inertial mass of an object is defined, operationally, by the
second law:

m=F/a,
where a is the acceleration of the mass m, caused by the force F. For a given particle, the mass
is constant; it has the same value in all inertial frames .

In Einsteinian Physics, the inertial mass, m, of an object depends on the speed of the
frame in which it is measured. If its mass is mo in its rest frame then its mass m in an inertial
frame moving at constant speed V is

m (the relativistic mass) = ynio (the rest mass) ,

where y is the same factor found in discussions of length contraction, and time dilation. The

validity of this equation has been demonstrated in numerous modem experiments.

The structure of atoms has been understood since the early 1930's. An atom consists of
a very small, positively charged nucleus, smrounded by electrons (negatively charged). The
nucleus contains protons (positively charged) and neutrons (electrically neutral) bound together
by the nuclear force. The diameter of a typical nucleus is less than 10~ 12 centimeters. The
electrons orbit the nucleus at distances that can be one hundred thousand times greater than the
size of the nucleus. The electrons are held in orbit by the electromagnetic force. The total
positive charge of the nucleus is exactly balanced by the total negative charge of the planetary
electrons, so that the atom is electrically neutral. An electron can be removed from an atom in
different ways, including ionization in an electric field and photo-ionization with light. A free
electron has the following accurately measured properties:

mass of the electron = 9.1083 x 10" 31 kilograms (kg),
and

electric charge of the electron = 1 .60206 x 10" 19 Coulomb. (C).
Here, "mass" refers to the mass measured in a frame of reference in which the electron is at
rest; it is the "rest mass".

In 1932, a particle was observed with the same mass as the electron and with a charge equal in
magnitude, but opposite in sign, to that of the electron. The particle was given the name

47
"positron"; it is the "anti-particle" of the electron. The concept of anti-particles was introduced
by Dirac, using purely theoretical arguments, a few years before the experimental observation
of the positron.

In the 1940's, it was found that an electron and a positron, when relatively at rest, may form a
"positronium atom" that consists of a bound state of an electron and a positron, orbiting about
their center-of-mass. Such an "atom" exists for a very short interval of time, namely 10" 10
seconds. It then spontaneously decays into two gamma-rays (high energy electromagnetic
radiation). The two gamma-rays are observed to travel back-to-back. (This observation is
consistent with the law of conservation of linear momentum; the total momentum is zero
before the decay (the particles are initially at rest), and therefore it must be zero after the decay).
Each gamma-ray has a measured energy of 0.51 1 MeV (Million electron-volts).
In Modem Physics, it is the custom to use the electron-volt (eV) as the unit of energy. An
electron-volt is the energy acquired by an electron when accelerated by a potential difference of
one volt. PictoriaUy,wehave:

(back-to-back)

+ - 0511 MeV 0.511 MeV
> * < — >

Matter -Anti-matter => Annihilation => Radiation

Electron-positron annihilation is a prime example of the conversion of matter into

electromagnetic radiation. (This is the basic process in PET scanning [Positron Emission

Tomography] in Nuclear Medicine ).

From the measured electron and positron masses, and the measured energies of the two

gamma-rays, we can obtain one of the most important results in our on-going quest for an

understanding of the laws of Nature, and the associated workings of the physical universe. We

begin by noting one of the standard results of Classical (Newtonian) Physics, namely, the

expression for the kinetic energy (energy of motion), E, of an object of mass m, moving with a

velocity v:

E = (L^mv 2 . (A result derived in standard texts on Mechanics).
We note that the ratio, energy/mass is

Em =(1/2)^, (the ratio is proportional to (velocity) 2 ).
We are therefore led to study the ratio:

gamma-ray energy/electron mass,
to obtain the resulting velocity-squared.
The gamma-ray energy is:

energy, E = 05 1 1 x 10 6 (eV) x 1 .602 x 10" 19 (Joule/eV)
= 0.8186 xlO" 13 Joule.
(A note on "units": in the Physical Sciences, units of measured quantities are often given in the
MKS system, in which lengths are given in meters, masses are given in kilograms, and time is

49
given in seconds. In this system, the unit of energy is the Joule (named after James Prescott
Joule, a Manchester brewer and distinguished scientist of the 19th-century)).
Our required conversion factor is: 1 electron-volt = 1 .602 x 10" 19 Joule.
In the MKS system, the ratio gamma-ray energy/ electron mass is therefore
E/m = 0.8186 x 10" 13 Joule / 9.1083 x 10" 31 kilogram,
= 0.89874 x 10 17 (meters/second) 2 ,
a velocity, squared.
We can find the velocity by taking the square root of the value of E/m, thus:

V(8.9874 x 10 16 ) = 2.9974 x 10 8 meters/second.
This is a truly remarkable result; it is the exact value of the measured velocity of light, always
written, c.
We therefore find that the ratio

E (gamma-ray)/m (electron) = c 2
or,

E=mc 2 .
This is Einstein's great equation that shows the equivalence of energy and mass. (Here, m is
the "relativistic mass" equal to yn\). It is important to note that Einstein derived this
fundamental relation using purely theoretical arguments, long before experiments were carried
out to verify its universal validity. The heat that we receive from the Sun originates in the
conversion of its central, highly compressed mass into radiant energy. A stretched spring has

more mass than an unsttetched spring, and a charged car battery has more mass than an

uncharged battery! In both cases, the potential energy stored in the systems has an equivalent

mass. We do not experience these effects because the mass changes are immeasurably small,

due to the 1/c 2 factor. However, in nuclear reactions that take place in nuclear reactors, or in

nuclear bombs, the mass (energy) differences are enormous, and certainly have observable

effects.

7. AN INTRODUCTION TO EINSTEINIAN GRAVITATION

7.1 The principle of equivalence

The term "mass" that appears in Newton's equation for the gravitational force between
two interacting masses refers to
"gravitational mass"; Newton's law should indicate this property of matter

F G = GWfrrP/r 1 , where M and m G are the gravitational masses of the
interacting objects, separated by a distance r.

The term "mass" that appears in Newton's equation of motion, F = ma, refers to
the "inertial mass"; Newton's equation of motion should indicate this property of matter:

F = nra, where m 1 is the inertial mass of the particle moving with an
acceleration a(r) in the gravitational field of the mass M* 3 .

Newton showed by experiment that the inertial mass of an object is equal to its
gravitational mass, m 1 = m G to an accuracy of 1 part in 10 3 . Recent experiments have shown
this equality to be true to an accuracy of 1 part in 10 12 . Newton therefore took the equations

51
F = GM G m G /r 2 = m I a
and used the condition m G = m 1 to obtain
a=GM G /r 2 .

Galileo had previously shown that objects made from different materials fall with the
same acceleration in the gravitational field at the surface of the Earth, a result that implies m G oc
m 1 . This is the Newtonian Principle of Equivalence.

Einstein used this Principle as a basis for a new Theory of Gravitation. He extended the
axioms of Special Relativity, that apply to field-free frames, to frames of reference in "free
fall". A freely falling frame must be in a state of unpowered motion in a uniform gravitational
field . The field region must be sufficiently small for there to be no measurable gradient in the
field throughout the region. The results of all experiments carried out in ideal freely falling
frames are therefore fully consistent with Special Relativity. All freely-falling observers
measure the speed of light to be c, its constant free-space value. It is not possible to carry out
experiments in ideal freely-falling frames that permit a distinction to be made between the
acceleration of local, freely-falling objects, and their motion in an equivalent external
gravitational field. As an immediate consequence of the extended Principle of Equivalence,
Einstein showed that a beam of light would be deflected from its straight path in a close
encounter with a sufficiently massive object. The observers would, themselves, be far
removed from the gravitational field of the massive object causing the deflection.

Einstein's original calculation of the deflection of light from a distant star, grazing the
Sun, as observed here on the Earth, included only those changes in time intervals that he had
predicted would occur in the near field of the Sun. His result turned out to be in error by
exactly a factor of two. He later obtained the "conecf ' value for the deflection by including in
the calculation the changes in spatial intervals caused by the gravitational field.
12 Rates of clocks in a gravitational field

Let a rocket be moving with constant acceleration a, in a frame of reference, F, far removed
from the Earth's gravitational field, and let the rocket be instantaneously at rest in F at time t =
0. Suppose that two similar clocks, 1 and 2, are attached to the rocket with 1 at therear end and
2 at the nose of the rocket. The clocks are separated by a distance L We can choose two light
sources, each with well-defined frequency, f^ as suitable clocks, fg is the frequency when the
rocket is at rest in an inertial frame in free space.

F (an inertial frame, no gravitational field)

constant acceleration, a, relative to F

Clocks at rest in rocket

acceleration begins at t >

Pulse of light emitted from 1 at t =

53
Let a pulse of light be emitted from the lower clock, 1, at time t = 0, when the rocket is
instantaneously at rest in F. This pulse reaches clock 2 after an interval of time t, (measured in
F) given by the standard equation for the distance traveled in time t:

ct = (^ + (l/2)at 2 ),
where ( l^at 2 is the extra distance that clock 2 moves in the interval t.
Therefore,

t=(£/c) + (a/2c)t 2 ,

*(^/c)if(at/2)«c
At time t, clock 2 moves with velocity equal to v = at« dllc, in F.

An observer at the position of clock 2 will conclude that the pulse of light coming from clock 1
had been emitted by a source moving downward with velocity v. The light is therefore
"Doppler-shifted", the frequency is given by the standard expression for the Doppler shift at
low speeds (v « c):

f«f [l-(v/c)]
= f [l-(a^/c 2 ].
The frequency f is therefore less than the frequency %. The light from clock 1 (below) is "red-
shifted". Conversely, light from the upper clock traveling down to the lower clock is measured
to have a higher frequency than the local clock 1 ; it is "blue-shifted".

The principle of equivalence states that the above situation, in a closed system, cannot

be distinguished by physical measurements, from that in which the rocket is at rest in a uniform
gravitational field. The field must produce an acceleration of magnitude lal, on all masses
placed in it.

y ' G is a non-accelerating frame with a uniform gravitational field present

Red shift

1 1

Blue shift

T T

Rocket at rest in G
Gravitational field

JVIassivebody

The light from the lower clock, reaching the upper clock will have a frequency lower than the
local clock, 2, by fog^/c 2 , (replacing lal by Igl), where g « 10 m/s 2 , the acceleration due to gravity

near the Earth. The light sources are at rest in G, and no oscillations of the pulses of light are
lost during transmission; we therefore conclude that, in a uniform gravitational field, factual
frequencies of the stationary clocks differ by fog^/c 2 . Now, g£ is the difference in the

"gravitational potential" between the two clocks. It is the convention to say that the upper
clock, 2, is at the higher potential in G . (Work must be done to lift the mass of clock 1 to dock

2 against the field).

55
Consider the case in which a light source of frequency 4 (corresponding to clock 1) is
situated on the surface of a star, and consider a similar light source on the Earth with a
frequency f E (corresponding to clock 2). Generalizing the above discussion to the case when
the two clocks are in varying gravitational fields, such that the difference in their potentials is
A(|),wefind

f^fsa+Ac^c 2 )
(gt = A(j), is the difference in gravitational potential of the clocks in
a constant field, g, when separated by I).

For a star that is much more massive than the Earth, A<j) is positive, therefore, f E > f s , or in terms
of wavelengths, X E and X s ,'k s >'k E . This means that the light coming from the distant star is
red-shifted compared with the light from a similar light source, at rest on the surface of the
Earth.

As another example, radioactive atoms with a well-defined "half-life" should decay
faster near clock 2 ( the upper clock) than near clock 1 . At the higher altitude (higher potential),
all physical processes go faster, and the frequency of light from above is higher than the
frequency of light from an identical clock below. Einstein's prediction was verified in a series
of accurate experiments, carried out in the late 1950's, using radioactive sources that were
placed at different heights near the surface of the Earth.

73 Gravity and photons

Throughout the 19th-century, the study of optical phenomena, such as the diffraction of
light by an object, demonstrated conclusively that light (electromagnetic in origin) behaves as a
wave. In 1900, Max Planck, analyzed the results of experimental studies of the characteristic
spectrum of electromagnetic radiation emerging from a hole in a heated cavity (so-called
"black-body radiation"). He found that current theory, that involved continuous frequencies in
the spectrum, could not explain the results. He did find that the main features of all black-body
spectra could be explained by making the radical assumption that the radiation consists of
discrete pulses of energy E proportional to the frequency, f . By fitting the data, he determined
the constant of proportionality, now called Planck's constant; it is always written h. The
present value is:

h = 6.626 x 10" 34 Joule-second in MKS units.
Planck's great discovery was the beginning of Quantum Physics.

In 1905, Einstein was the first to apply Planck's new idea to another branch of Physics,
namely, the Photoelectric Effect. Again, current theories could not explain the results. Einstein
argued that discrete pulses of electromagnetic energy behave like localized particles, carrying
energy E = hf and momentum p = E/c. These particles interact with tiny electrons in the
surface of metals, and eject electrons in a Newtonian-like way. He wrote

Ep H =nfp H andEp H =p PH c
The rest mass of the photon is zero. (Its energy is all kinetic).

57
If, under certain circumstances, photons behave like particles, we are led to ask: are
photons affected by gravity? We have

ripfj = ir^ cr — hr PH ,
or

1%/ = Eppj/c 2 = hfpjj/c 2 .
By the Principle of Equivalence, inertial mass is equivalent to gravitational mass, therefore
Einstein proposed that abeam of light (photons) should be deflected in a gravitational field, just
as if it were a beam of particles. (It is worth noting that Newton considered light to consist of
particles; he did not discuss the properties of his particles. In the early 1800's, Soldner actually
calculated the deflection of a beam of "light-particles" in the presence of a massive object!
Einstein was not aware of this earlier work).

Let us consider a photon of initial frequency f s , emitted by a massive star of mass M s ,
and radius R. The gravitational potential energy, V, of a mass m at the surface of the star, is
given by a standard result of Newton's Theory of Gravitation; it is

V(surface) = - GM s m/R.
It is inversely proportional to the radius of the star. The negative sign signifies that the
gravitational interaction between Mj and m is always attractive.

Following Einstein, we can write the potential energy of the photon of "mass" hfpf/c 2 at
the surface as

V (surface) = - (GMs/PvXhfpn/c 2 ).

The total energy of the photon, E^^ is the sum of its kinetic and potential energy:
E rorA L=nf PH OTAR + BGM^^/Rc 2 ,
=hf PH srAR (l-GM s /Rc 2 ).
At very large distances from the star, at the Earth, for example, the photon is essentially beyond
the gravitational pull of the star. Its total energy remains unchanged (conservation of energy).
At the surface of the Earth the photon has an energy that is entirely electromagnetic (since its
potential energy in the "weak" field of the Earth is negligible compared with that in the
gravitational field of the star), therefore

so that

M m BAKm = hfpH CTAR (1 - GM S /Rc 2 )

f PH EAR ™/fpH SrAR =l-GM s /Rc 2 ,

and

Afif - (f m STm - fpH^/fpH™* = GMs/Rc 2 .
We see that the photon on reaching the Earth has less total energy than it had on leaving
the star. It therefore has a lower frequency at the Earth. If the photon is in the optical region, it
is shifted towards the red-end of the spectrum. This is the gravitational red-shift. (It is quite
different from the red-shift associated with Special Relativity)
Schematically, we have:

STAR

To Earth W™

Mass Ms / Blue light emitted

Light red-shifted

Massive Star

far from field of star

74 Black holes

In 1784, a remarkable paper was published in the Philosophical Transactions of the
Royal Society of London, written by the Rev. J. Michell. It contained the following discussion:

To escape to an infinite distance from the surface of a star of mass M and radius R, an
object of mass m must have an initial velocity v given by the energy condition:
initial kinetic energy of mass > potential energy at surface of star,

This means that

(l/2)mv 2 > GMm/R (A Newtonian expression).

v >V(2GM/R).

Escape is possible only when the initial velocity is greater than (2GM/R) 1

,1/2

On the Earth, v > 25 ,000 miles/hour.

For a star of given mass M, the escape velocity increases as its radius decreases. Michell
considered the case in which the escape velocity v reaches a value c, the speed of light. In this
limit, the radius becomes

R UMrr = 2GM/c 2
He argued that light would not be able to escape from a compact star of mass M with a radius
less than R^m^; the star would become invisible. In modem terminology, it is ablack hole.

Using the language of Einstein, we would say that the curvature of space-time in the
immediate vicinity of the compact star is so severe that the time taken for light to emerge from
the star becomes infinite. The radius 2GM/C 2 is known as the Schwarzschild radius; he was
the first to obtain a particular solution of the Einstein equations of General Relativity. The
analysis given by Michell, centuries ago, was necessarily limited by the theoretical knowledge
of his day. For example, his use of a non-relativistic expression for the kinetic energy (mv 2 ^)
is now known to require modification when dealing with objects that move at speeds close to
c. Nonetheless, he obtained an answer that turned out to be essentially correct. His use of a
theoretical argument based on the conservation of energy was not a standard procedure in
Physics until much later.

A star that is 1 .4 times more massive than our Sun, has a Schwarzschild radius of only
2km and a density of 10 20 kg/m 3 . This is far greater than the density of an atomic nucleus. For
more compact stars (R LIMir < 1.4 M SUN ), the gravitational self-attraction leads inevitably to its
collapse to a "point'.

61
Studies of the X-ray source Cygnus X- 1 indicate that it is a member of a binary system,
the other member being a massive "blue supergiant". There is evidence for the flow of matter
from the massive optical star to the X-ray source, with an accretion disc around the center of
the X-ray source. The X-rays could not be coming from the blue supergiant because it is too
cold. Models of this system, coupled with on-going observations, are consistent with the
conjecture that a black hole is at the center of Cygnus X-l . Several other good candidates for
black holes have been observed in recent studies of binary systems. The detection of X-rays
from distant objects has become possible only with the advent of satellite-borne equipment.

I have discussed some of the great contributions made by Einstein to our understanding
of the fundamental processes that govern the workings of our world, and the universe, beyond.
He was a true genius, he was a visionary, and he was a man of peace.

62
Appendix

The following material presents the main ideas of Einstein's Special Relativity in a mathematical
form. It is written for those with a flair for Mathematics.
Al. Some useful mathematics: transformations and matrices

Let a point P[x, y] in a Cartesian frame be rotated about the origin through an angle of

90°; let the new position be labeled P[x', y']

-x

P'[x',y'] \

^N^

, P[X

►

We see that the new coordinates are related to the old coordinates as follows:

x' (new) = -y (old)

and

y' (new) = +x (old)

where we have written the x's and y's in different columns for reasons that will become clear,

later.

Consider a stretching of the material of the plane such that all x-values are doubled and all y-

values are tripled:

3y-

P'[x',y'] = P'[2x,3y]

P[x,y]

x 2x

The old coordinates are related to the new coordinates by the equations

Y= 3y

and

x'=2x

Consider a more complicated transformation in which the new values are combinations of the

old values, for example, let

x' = lx + 3y

and

y = 3x + ly

We can see what this transformation does by putting in a few definite values for the

coordinates:

[0,0] -[0,0]

[1,0] — [1.1 +3.0,3.1 + 1.0] = [1,3]

[2, 0] — [1 2 + 3.0, 3.2 + 1 .0] = [2, 6]

[0,1] -»[10 + 3.1, 3.0 +1.1] = [3,1]

[0, 2] -* [1 JO + 32, 3.0 + 1 2] = [6, 2]

[1,1] -»[1.1 +3.1,3.1 + 1.1] = [4,4}

[1,2] -»[1.1 +32,3.1 + 1.2] = [7,5]

[2,2] -^[12 + 32,32 + 12] = [8,8]

[2, 1] -* [12 + 3.1,32+1.1] = [5,7]

and so on.

Some of these changes are shown below

Ne^

va>

:es;

and

gri(

1-lin

esa

reo

blic

l #

„/

This is a particular example of the more general transformation

x' = ax+by

and

y' = ex + dy

where a, b, c, and d are real numbers.

In the above examples, we see that each transformation is characterized by the values of the

coefficients, a, b, c, and d:

For the rotation through 90°:

a=0,b = -l,c=l,andd = 0;

for the 2x3 stretch.

a=2,b=0,c=0,andd = 3;

and for the more general transformation:

a=l,b = 3,c = 3,andd=l.

In the 1840's, Cayley recognized the key role of the coefficients in characterizing the
transformation of a coordinate pair [x, y] into the pair [x', y']. He therefore "separated them
out", writing the pair of equations in column-form, thus:

r -\

■>

r ~\

—

vy^

This is a single equation that represents the original two equations. We can write it in the

symbolic form:

F = MP,

which means that the point P with coordinates x, y (written as a column) is changed into the

point P' with coordinates x', y' by the operation of the 2 x 2 matrix operator M.

The matrix Mis

a b
M =

The algebraic rule for carrying out the "matrix multiplication" is obtained directly by noting
that the single symbolic equation is the equivalent of the two original equations. We must
therefore have

x' = a times x + b times y
and

y = c times x + d times y.
We multiply rows of the matrixby columns of the coordinates, in the conect order.
2x2 matrix operators will be seen to play a crucial role in Einstein's Special Theory of
Relativity.
A2. Galilean-Newtonian relativity revisited

The idea of matrix operators provides us with a useful way of looking at the equations
of classical relativity, discussed previously. Recall the two basic equations:

f = t
and

x' = x-Vt .
where, the event E[t, x] in the F-frame has been transformed into the event E'[t', x'] in the F-
frame. We can write these two equations as ^single matrix operator equation as follows

"1 r ^
1 t

-v i JLx,

or, symbolically

where

E' = GE,

G =

.-v 1

, the matrix of the Galilean transformation.

If we transform E -* E' under the operation G, we can undo the transformation by
carrying out the inverse operation, written G" 1 , that transforms E' -» E, by reversing the
direction of the relative velocity:

t = f

and

x=x' + Vf

or, written as a matrix equation:

1 t'

xj ,+V l^vX

where

G =

+V 1

is the inverse operator of the Galilean transformation. Because G l undoes

the effect of G, we have

G G = "do nothing" = I , the identity operator,

where

1 =

(0 1,

We can illustrate the space-time path of a point moving with respect to the F- and F-
frames on the same graph, as follows

x-axis

0,0'

E[t,x]andE'[t',x']

D = Vt

t'-axis

(the world line
ofO' relative to O)

The origins of F and F are chosen to be coincident at t = f = 0. O' moves to the right
with constant speed V, and therefore travels a distance D = Vt in time t. The t - axis is the
world line of O' in the F-frame. Every point in this space-time geometry obeys the relation

x' = x - Vt; the F-frame is therefore represented by a semi-oblique coordinate system. The

69
characteristic feature of Galilean-Newtonian space-time is the coincidence of the x-x'- axes.
Note that the time intervals, t, t in F and F are numerically the same (Newton's "absolute
time"), and therefore a new time scale must be chosen for the oblique axis, because ^lengths
along the time-axis, corresponding to the times t, f of the event E, E' are different.
A3. Is the geometry of space-time Pythagorean?

Pythagoras' Theorem is of primary importance in the geometry of space. The theorem
is a consequence of the invariance properties of lengths and angles under the operations of
translations and rotations. We are therefore led to ask the question - do invariants of space-
time geometry exist under the operation of the Galilean transformation and, if they do, what are
they? We can address this question by making a simple, direct calculation, as follows. The
basic equations that relate space-time measurements in two inertial frames moving with
relative speed V are

x' = x-Vt and t' = t.
We are interested in quantities of the form x 2 + 1 2 and x' 2 + 1' 2 . These forms are inconsistent,
however, because the "dimensions" of the terms are not the same; x, x' have dimensions of
"length" and t, f have dimensions of "time". This inconsistency can be dealt with by
introducing two quantities k, k' that have dimensions "length/time" (speed), so that the
equations become

x' = x - Vt (all lengths) and Yt = kt (all lengths).
(Note that kt is the distance traveled in a time t at a constant speed k) . We now find

x' 2 = (x - Vt) 2 = x 2 - 2xVt + V¥, and k¥' 2 = k¥,

so that

x' 2 + k'¥ 2 = x 2 - 2xVt + V¥ + k¥.

* x 2 + k¥ unless V = (no motion!).
Relative events in an semi-oblique space-time geometry therefore transform under the Galilean
operator in a non-Pythagorean way.
A4. Einstein's space-time symmetry: the Lorentz transformation

We have seen that the classical equations relating the events E and E' are
E' = GE, and the inverse E = G _1 E' where

G =

and G" 1 =

-V 1

These equations are connected by the substitution V <-» -V; this is an algebraic statement of
the Newtonian Principle of Relativity. Einstein incorporated this principle in his theory (has first
postulate), broadening its scope to include all physical phenomena, and not simply the motion
of mechanical objects. He also retained the linearity of the classical equations in the absence of
any evidence to the contrary. (Equispaced intervals of time and distance in one inertial frame
remain equispaced in any other inertial frame). He therefore symmetrized the space-time
equations (by putting space and time on equal footings) as follows:

1 -V t

-V replaces 0, to symmetrize the matrix

-V 1

Note, however, the inconsistency in the dimensions of the time-equation that has now been

introduced:

t'=t-Vx.

The term Vx has dimensions of [L] 2 /[T] , and not [T] . This can be corrected by introducing the
invariant speed of light, c (Einstein's second postulate, consistent with the result of the
Michelson-Morley experiment):

ct' = ct - Vx/c (c' = c, in all inertial frames)
so that all terms now have dimensions of length, (ct is the distance that light travels in a time t)
Einstein went further, and introduced a dimensionless quantity y instead of the scaling
factor of unity that appears in the Galilean equations of space-time. (What is the number "1"
doing in a theory of space-time?). This factor must be consistent with all observations. The
equations then become

ct'= Y ct ~ P^

x' = — (3yct + yx , where |3=V/c.

These can be written

E =LE,

72
where

L =

Y -PY

-Py y

and

E = [ct,x].

L is the operator of the Lorentz transformation. (First obtained by Lorentz, it is the
transformation that leaves Maxwell's equations of electromagnetism unchanged in form
between inertia! frames).

The inverse equation is E = L E' , where

L =

Y PY

Py y

This is the inverse Lorentz transformation, obtained fromL by changing |3 -* -|3 (V -» -V); it
has the effect of undoing the transformation L. We can therefore write

LL _1 = I, the identity.
Carrying out the matrix multiplication, and equating elements gives

Y 2 -|3Y=1

therefore,

Y = 1A/(1 - 13 2 ) (taking the positive root).

As V -* 0, |3 -* and therefore y -* 1 ; this represents the classical limit in which the Galilean
transformation is, for all practical purposes, valid. In particular, time intervals have the same

73
measured values in all Galilean frames of reference, and acceleration is the single Galilean
invariant.
A5. The invariant interval

Previously, it was shown that the space-time of Galileo and Newton is not Pythagorean
under G. We now ask the question: is Einsteinian space-time Pythagorean under L ? Direct
calculation leads to

(ct) 2 + x 2 = ^(1 + 13 2 )(0 2 + 40yVct'
+Y 2 (l + |3 2 )x' 2
*(ct0 2 + x' 2 if|3>0.
Note, however, that the difference of squares is an invariant:

(ctf-x 2 = (ct'f-x 2
because

Y 2 (l-(3 2 ) = 1.
Space-time is said to be pseudo-Euclidean. The "difference of squares" is the characteristic
feature of Nature's space-time. The "minus" sign makes no sense when we try and relate it to
our everyday experience of geometry. The importance of Einstein's "free invention of the
human mind" is clearly evident in this discussion.

The geometry of the Lorentz transformation, L, between two inertial frames involves
oblique coordinates, as follows:

x-axisf tan ! |3

Common O, O

'[cV,X]

ct-axis

The symmetry of space-time means that the ct - axis and the x - axis fold through equal angles.
Note that when the relative velocity of the frames is equal to the speed of light, c, the folding
angle is 45°, and the space-time axes coalesce.
A6. The relativity of simultaneity: the significance of oblique axes

Consider two sources of light, 1 and 2, and a point M midway between them. Let F^
denote the event "flash of light leaves 1", and F^ denote the event "flash of light leaves 2". The
events E l and L^ are simultaneous if the flashes of light from 1 and 2 reach M at the same time.
The oblique coordinate system that relates events in one inertial frame to the same events in a
second (moving) inertial frame shows, in a most direct way, that two events observed to be
coincident in one inertial frame are not observed to be coincident in a second inertial frame

(moving with a constant relative velocity , V, in standard geometry) . Two events E, [ctj , xj and
EJct^ xj , are observed in a frame, F. Let them be coincident in F, so that \ = ^ = t, (say). The
two events are shown in the following diagram:

x-axis

F-frame

Ctj = c^ = ct

Coincident events in F

ct-axis

Consider the same two events as measured in another inertial frame, F , moving at constant
velocity V along the common positive x - x' axis. In F, the two events are labeled E/[ct/, x/]
and E/fctj' , x 2 ' ] . Because F is moving at constant velocity +V relative to F, the space-time
axes of F are folded inwards through angles tan _1 (V/c) relative to the F axes, as shown. The

events E/ and E/ can be displayed in the F - frame:

inclined at tan '(V/c) relative to the x - axis of F

']inF

Not coincident in F

ct' - axis inclined at tan ^/c)

relative to the ct - axis of F

t/< tj' in F ctj = c^ (Ej and F^ are coincident in F)

We therefore see that, for all values of the relative velocity V > 0, the events E/ and E/ as
measured in F are not coincident; E/ occurs before E/ .

(If the sign of the relative velocity is reversed, the axes fold outwards through equal angles).
A7. Length contraction: the Lorentz transformation in action

The measurement of the length of arod involves comparing the two ends of the rod
with marks on a standard ruler, or some equivalent device. If the object to be measured, and
the ruler, are at rest in our frame of reference then it does not matter when the two end-positions

are determined - the "length" is clearly-defined. If, however, the rod is in motion, the meaning

77
of its length must be reconsidered. The positions of the ends of the rod relative to the standard
ruler must be "measured at the same time" in its frame of reference.

Consider a rigid rod at rest on the x'-axis of an inertial reference frame F". Because it is
at rest, it does not matter when its end-points x{ and x 2 ' are measured to give the rest-, or
proper-length of the rod, L^ = x 2 " - x{.

Consider the same rod observed in an inertial reference frame F that is moving with constant
velocity -V with its x-axis parallel to the x'-axis. We wish to determine the length of the
moving rod; We require the length L = x 2 - x l according to the observers in F. This means
that the observers in F must measure x { and x 2 at the same time in their reference frame. The
events in the two reference frames F, and F are related by the spatial part of the Lorentz
transformation:

x'=-|3yct+Yx
and therefore

x 2 ' -x{ = -Pyc^ - tj + y(x 2 - x t ).
where

|3 = V/candY=l/V(l-|3 2 ).
Since we require the length (x 2 - Xj) in F to be measured at the same time in F, we must have ^
- tj = 0, and therefore

V = X 2 -V = Y( X 2~ X l) .

^(at nest) = yh (moving) .
The length of a moving rod, L, is therefore less than the length of the same rod measured at
rest, Lq ,because y > 1 .
A8. Time dilation: a formal approach

Consider a single clock at rest at the origin of an inertial frame F, and a set of
synchronized clocks at Xq, x 1? x 2 , ... on the x-axis of another inertial frame F. Let V move at
constant velocity +V relative to F, along the common x -, x'- axis. Let the clocks at x^ and \
be synchronized to read ^ and t^ at the instant that they coincide in space. A proper time
interval is defined to be the time between two events measured in an inertial frame in which
they occur at the someplace. The time part of the Lorentz transformation can be used to relate
an interval of time measured on the single clock in the F frame, and the same interval of time
measured on the set of synchronized clocks at rest in the F frame. We have

ct=ycf + |3yx'
or

c(t 2 -t 1 )= ycCV-V) + (3y(x/- Xi 0.
There is no separation between a single clock and itself, therefore x/ - x{ = 0, so that

c(t2 - ^(moving) = Yc(t/ - t{)(ai rest) ,
or

cAt (moving) = ycAt' (at rest).
Therefore, because y > 1 , a moving clock runs more slowly than a clock at rest.

79
A9. Relativistic mass, momentum, and energy

The scalar product of a vector A with components [^ , aj and a vector B with
components [b^bj is

AB = ajbj + a 2 b 2 .
In geometry, AB is an invariant under rotations and translations of the coordinate
system.

In space-time, Nature prescribes the differences-of-squares as the invariant under the
Lorentz transformation that relates measurements in one inertial frame to measurements in
another. For two events, E^ct, x] and FJct, -x] , the scalar product is
E 1 -E 2 =[ct,x].[ct,-x]

= (ct) 2 - x 2 = invariant in a space-time geometry,
where we have chosen the direction of F^ to be opposite to that of E^thereby providing the
necessary negative sign in the invariant.
In terms of finite differences of time and distance, we obtain

(cAt) 2 - (Ax) 2 = (cAx) 2 = invariant,
where Ax is the proper time interval. It is related to At by the equation

At=yAT.
In Newtonian Mechanics, the quantity momentum, the product of the mass of an object
and its velocity, plays a key role. In Einsteinian Mechanics, velocity, mass, momentum and

kinetic energy are redefined. These basic changes are a direct consequence of toe replacement
of Newton's absolute time interval, At N , by the Einstein's velocity-dependent interval
At E =yAx.
The Newtonian momentum ^ - m N v N = n^Ax/A^ is replaced by the Einsteinian
momentum

Pe + ■ moV E = moA[ct, x]/Ax
= mJcAt/Ax, Ax/Ax]
= mofyc, (Ax/At)(At/Ax]
=mo[Yc,YV N ].
We now introduce the vector in which the direction of the x-component is reversed, giving

Forming the scalar product, we obtain

p E + p E ~ = m 2 (Y 2 c 2 -Y 2 v N 2 )
= m 2 c 2 ,

because v E + -v E ~ = c 2 .

Multiplying throughout by c 2 , and rearranging, we find

mo 2 c 4 = fm^c 4 - yWc^n 2 •
We see that y is a number and therefore y multiplied by the rest mass n^ is a mass; let us
therefore denote it by m:

m = ymo, the relativistic mass.

81
We can then write

Hcf = (mc 2 ) 2 -(CPE) 2 .
The quantity me 2 has dimensions of energy, let us therefore denote it by the symbol E, so that

E = me 2 , Einstein's great equation.
The equivalence of mass and energy is seen to appear in a natural way in our search for the
invariants of Nature.
The term involving mo is the rest energy, Eq,

Eo^nV 2 .
We therefore obtain

E^ 2 = E 2 - (PeP) 2 = E' 2 - (Pe'c) 2 , in any other inertial frame.

It is ^fundamental invariant of relativistic particle dynamics.

This invariant includes those particles with zero rest mass. For a photon of total energy

EpH and momentum p^, we have

= E PH 2 -(p PH c) 2 ,

and therefore

No violations of Einstein 's Theory of Special Relativity have been found in any tests of
the theory that have been carried to this day.

82
Bibliography

The following books are written in a style that requires little or no Mathematics:
Calder,N., Einstein's Universe,The Viking Press,New York (1979).
Davies, P. C. W., Space and Time in the Modern Universe,

Cambridge University Press, Cambridge (1977).
The following books are mathematical in style; they are listed in increasing level of
mathematical sophistication:
Casper, Barry M., and Noer, Richard J. Revolutions in Physics,

W. W. Norton & Company Inc., New York (1972).
Bom, M., The Special Theory of Relativity, Dover, New York (1962).
French, A. P., Special Relativity, W. W. Norton & Company, Inc.

New York (1968).
Rosser, W. G. V., Introduction to Special Relativity, Butterworth & Co. Ltd.

London (1967).
Feynman, R. P., Leighton, R. B ., and Sands, M., The Feynman Lectures on

Physics, Addison-Wesley Publishing Company, Reading, MA (1964).
Rindler, W '., Introduction to Special Relativity, Oxford University Press,

Oxford, 2nd ed. (1991).

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